We showed in [1]}, [2]}
that the Liouville equation (REF ), used to evolve
the probability distribution defining the average in
Eq. (REF ), implies the equation of motion for the moment
hierarchy
\({\frac{d}{d\tau } {\Phi }_k =\dot{\prod _p}\left[\sum _{j_p = 0}^{k_p}\left(\begin{array}{c}k_p \\ j_p\end{array}\right)Y_p^{\underline{j_p}} \,e^{{\left( k - j \right)}_p \, \partial / \partial Y_p}\right]{\mathbb {A}}_{\kappa }\left<{\Psi }_{Y} \! \left( {\rm n}\right)\right>} &\nonumber \\& =\sum _{j_1 = 0}^{k_1}\left(\begin{array}{c}k_1 \\ j_1\end{array}\right) \ldots \sum _{j_P = 0}^{k_P}\left(\begin{array}{c}k_P \\ j_P\end{array}\right)\left[\dot{\prod _p}Y_p^{\underline{j_p}} \,\right]{\mathbb {A}}_{\kappa }\left<{\Psi }_{Y + \left( k - j \right)} \! \left( {\rm n}\right)\right>\nonumber \\& \equiv \sum _{k^{\prime }}{\Lambda }_{k k^{\prime }}{\Phi }_{k^{\prime }} .\) 
